Understanding Doubly-Infinitary Distributive Categories and Cartesian Closedness

Core Concepts

Free doubly-infinitary distributive categories are cartesian closed.

Abstract

The article explores the concept of doubly-infinitary distributive categories, showing their relation to established notions like infinitary distributiveness and cartesian closedness. It presents examples, counterexamples, and theoretical results regarding these categories. The main focus is on the comparison between doubly-infinitary distributivity and cartesian closedness in various mathematical contexts.

Stats

Free doubly-infinitary distributive categories are shown to be cartesian closed. Quasi-Borel spaces provide a counterexample where a category is cartesian closed but not doubly-infinitarily distributive. The category FinSet of finite sets lacks infinite products and coproducts, making it not doubly-infinitarily distributive.

Quotes

"The category Set is doubly-infinitary distributive." "LocConTop is doubly-infinitaly distributive but not cartesian closed." "Cartesian closed categories are not necessarily doubly-infinitaly distributive."

Key Insights Distilled From

Free Doubly-Infinitary Distributive Categories are Cartesian Closed

by Fern... at arxiv.org 03-18-2024

https://arxiv.org/pdf/2403.10447.pdf

Deeper Inquiries

What implications do the examples of quasi-Borel spaces and FinSet have for the relationship between cartesian closure and double-infinitary distribution

The examples of quasi-Borel spaces and FinSet provide valuable insights into the relationship between cartesian closure and doubly-infinitary distribution. In the case of quasi-Borel spaces, we see a scenario where a category can be both cartesian closed and infinitary distributive but still fail to be doubly-infinitary distributive. This highlights that while cartesian closed categories are necessarily infinitary distributive, they may not always exhibit double-infinitary distribution. On the other hand, considering FinSet, which lacks infinite products and coproducts, it serves as a counterexample demonstrating that a category without these properties cannot be doubly-infiniarily distributive.

How does the concept of connected objects contribute to understanding doubly-infinitary distributivity

The concept of connected objects plays a crucial role in understanding doubly-infinitary distributivity. When every object in a category is expressible as a coproduct of connected objects with products preserved within those connected components, then the entire category becomes doubly-infinitarily distributive. This insight allows us to identify conditions under which categories exhibit this unique property by focusing on the connectivity structure within the objects present.

In what ways can the study of free completion under coproducts enhance our understanding of categorical structures

Studying free completion under coproducts provides significant enhancements to our understanding of categorical structures. By exploring concepts such as free completion pseudomonads arising from canonical (pseudo)distributive laws between products and coproducts, we gain insights into how limits and colimits interact within categories with specific properties like double-infinitary distribution. Additionally, examining pseudoalgebras related to free completions helps us characterize categories based on their product-preserving or coproduct-preserving behaviors, leading to deeper comprehension of their structural properties and relationships between different categorical structures.